If you are coming from an electronics background may I suggest the operating systems course, this fills in the blanks of how do logic gates on silicon end up providing an environment to run applications.
I started from an electronic (telecoms/communications) background before entering computers so I always wanted to know how things really worked, eg from the metal upwards.
We did our database course on SQL and never touched a computer. It seemed logical but abstract.
There are many people here who come from an Electrical/Electronics or related dicipline, including all three of the Admins, so for a start you should not feel that this is anything of a disadvantage in getting where you want to go. Any knowledge gained in either area will be useful, maybe not directly, but will stand to you in the long term.
As for the specifics of this case, cbkihong has already indicated that most modern websites are in fact web "applications" sitting in front of databases of varying size, from the small site where someone has their own blog and nothing else, to huge corporate sites to online shops. In order to be an effective web designer you will need to know how to interact efficiently with databases in order to develop effective websites, so my advice would be to take the database course.
I came from a Hotel Management 4 years degree background. I slipped in the simple admin area(back-up, users & passwords), next thing I am looking at spreadsheets, which took me to databases and forms, that forced me to install it on different operating systems, which got me to study them, how they work, how they are built, that had me do low level programming, which ended up in extensive readings about electronics and electric structures and circuits.
All self tought, from official (and non-official) used university books. Now, after 12 years of touching a keyboard for the first time ever, and 10 years in IT hall ways and rooms, I am a novice UNIX guy, with Linux and Solaris, and a couple of Cisco certificates.
Where do I recommend you to start? I think that all the ways lead to Rome.
Do what you enjoy doing, if it is web, then let it be. Just have fun doing it. If you don't, trust me, it will be a living nightmare. That is why they'll pay you the big bucks, because they do not enjoy doing it, and you do.